Business Mathematics: Interpreting Data - Lesson 5.3 PDF

Unit 5: Presentation and Analysis of Business Data Lesson 5.3 Interpreting Data: Measures of Central Tendency Contents Introduction 1 Learning Objectives 2 Quick Look...

Unit 5: Presentation and Analysis of Business Data Lesson 5.3 Interpreting Data: Measures of Central Tendency Contents Introduction 1 Learning Objectives 2 Quick Look 3 Learn the Basics 4 Mean 5 Ungrouped Data 5 Grouped Data 7 Median 10 Ungrouped Data 10 Grouped Data 12 Mode 15 Ungrouped Data 15 Grouped Data 16 Business-Related Problems 19 Case Study 23 Keep in Mind 24 Try This 25 Practice Your Skills 26 Challenge Yourself 27 Photo Credit 29 Bibliography 29 Unit 5: Presentation and Analysis of Business Data Lesson 5.3 Interpreting Data: Measures of Central Tendency Introduction Suppose you have a refreshment stall business. One way to better understand how your business performs is to keep in mind the average income. To determine your average daily, weekly, or monthly income, you can use measures of central tendency. In this lesson, you will apply the concepts of mean, median, and mode in understanding averages in the context of business performance and operations. Also, you will solve business-related problems by applying the different measures of central tendency. 5.3. Interpreting Data: Measures of Central Tendency 1 Unit 5: Presentation and Analysis of Business Data Learning Objectives DepEd Competency At the end of this lesson, you should be able to Analyze and interpret the data presented in the do the following: table using measures of central tendency and Differentiate the various measures of variability and tests of significant differences (ABM_BM11PAD-IIh-5). central tendency. Calculate and interpret measures of central tendency from tabular data. Solve business-related problems involving measures of central tendency. 5.3. Interpreting Data: Measures of Central Tendency 2 Unit 5: Presentation and Analysis of Business Data Quick Look Average Income Dan recorded his daily income for his refreshment stall business. From Monday through Sunday, his incomes were ₱8,050, ₱8,100, ₱8,550, ₱7,895, ₱8,700, ₱8,910, and ₱9,050, respectively. If he targets an average daily income of more than ₱8,300, determine whether this goal was achieved or not. Solution Step 1: Identify what is required in the problem. You are asked to determine if the target average daily income (I) of more than ₱8,300 was achieved or not. Step 2: Identify the given in the problem. ₱8,050 Monday (m) ₱8,100 Tuesday (t) ₱8 550 Wednesday (w) ₱7,895 Thursday (h) ₱8,700 Friday (f) ₱8,910 Saturday (s) ₱9,050 Sunday (u) ₱8,300 Target average daily income (I) Step 3: Write the working equation. Step 4: Substitute the given values. Step 5: Find the answer. 5.3. Interpreting Data: Measures of Central Tendency 3 Unit 5: Presentation and Analysis of Business Data Dan achieved his goal. The average daily income is ₱8,465, surpassing the target daily income of more than ₱8,300. Questions to Ponder 1. What is an average? Are there any forms of averages? __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ 2. How are mean, median, and mode interpreted? __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ 3. How do you apply the concepts of mean, median, and mode to business transactions? __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ Learn the Basics When you hear about a business’s average income, you generally think of a number near the middle of the income scale. In general, that is right — the average is the most common value. There are three ways to calculate the average. What are these? A measure of central tendency refers to a single score or value that can best describe a given set of observations. It can be in the form of mean, median, or mode. Each of the measures of central tendency is used depending on certain factors. 5.3. Interpreting Data: Measures of Central Tendency 4 Unit 5: Presentation and Analysis of Business Data Essential Question What business insights can be generated from measuring the central tendency? Mean Mean is one of the measures of central tendency, which can be obtained by determining the average set of scores. You can use this method to have a reliable value in describing a set of scores. The data can be presented as either ungrouped or grouped. Ungrouped Data To compute the mean of an ungrouped data, you must add all the data in a group of values and divide by the number of data present. To represent in an equation: where, x̄ = mean n = the number of values 5.3. Interpreting Data: Measures of Central Tendency 5 Unit 5: Presentation and Analysis of Business Data Closer Look Example 1 What is the mean of 95, 99, 89, 85, 90, 88, and 94? Solution: The mean value of the data set is 91.43 Example 2 Your monthly electric bill for the first five months are the following: January = ₱1,805 February = ₱1,612 March = ₱1,988 April = ₱2,004 May = ₱1,846 a. What is your average monthly electric bill? b. What does it mean? Solution: a. What is your average monthly electric bill? 5.3. Interpreting Data: Measures of Central Tendency 6 Unit 5: Presentation and Analysis of Business Data Therefore, your average monthly electric bill is ₱1 851. b. Interpret your answer in (a). A mean value of ₱1,851 indicates that such a value is the central or typical value in a data set. Grouped Data For grouped data, you can compute the mean by following these steps. 1. Compute the midpoint of each of the classes. 2. Multiply each of the frequencies to their corresponding midpoints and get the sum. 3. Divide the sum by the number of values. The formula in solving for the mean of a grouped data is: where, x̄ = mean f = frequency of each class x = midpoint of each class n = number of values Properties of the mean (x̄): 1. It may or may not be an actual score in the distribution. 2. All scores affect the mean value. 3. It is easily affected by outliers. 5.3. Interpreting Data: Measures of Central Tendency 7 Unit 5: Presentation and Analysis of Business Data Closer Look Example 1 The table below shows the ages of customers who visit the candy store. What is the mean value? Age Frequency 3-9 11 10-16 4 17-23 2 24-30 2 31-37 1 n 20 Solution: 1. Compute the midpoint of each of the classes. To obtain the midpoint, get the average value of the upper and lower classes. Age Frequency (f) Midpoint (x) 3-9 11 6 10-16 4 13 17-23 2 20 24-30 2 27 31-37 1 34 5.3. Interpreting Data: Measures of Central Tendency 8 Unit 5: Presentation and Analysis of Business Data 2. Multiply each of the frequencies to its corresponding midpoints and get the sum. Age Frequency (f) Midpoint (x) fx 3-9 11 6 66 10-16 4 13 52 17-23 2 20 40 24-30 2 27 54 31-37 1 34 34 3. Divide the sum by the number of values. Therefore, the average age of customers who visit the candy store is 12.3 years old. Store 5.3. Interpreting Data: Measures of Central Tendency 9 Unit 5: Presentation and Analysis of Business Data Check Your Progress You recorded the prices of milkfish in different markets in your town. The costs per kilogram are ₱180, ₱190, ₱180, ₱200, ₱190, ₱195, ₱170, and ₱185. What is the average price of the milkfish? Median The median is the middlemost score of a set of distributions if arranged in an ascending or a descending order. This value divides the set of data into two equal sets. This measure of central tendency is used when there are extreme values. Ungrouped Data To obtain median of an ungrouped data, you should follow these steps: 1. Arrange the data in an ascending or a descending order. 2. Determine the middlemost value. 2.1. If the number of values is odd, the median is the middlemost value. 2.2. If the number of values is even, the median is the average of the two middlemost values. 5.3. Interpreting Data: Measures of Central Tendency 10 Unit 5: Presentation and Analysis of Business Data Closer Look Example 1 What is the median of these values: 5,16,15,2,17,19,and 10? Solution: Step 1: Arrange the data in an ascending or a descending order. This is arranged in descending order: 19, 17, 16, 15, 10, 5, 2 Step 2: Determine the middlemost value. Since the number of values is 7, which is odd, the median is 15. The median value of 5,16,15,2,17,19,and 10 is 15. Example 2 A customer observes the changes in oil price for the past weeks. From week 1 to 8, the prices are ₱50.5, ₱52.8, ₱49.69, ₱50.45, ₱54.56, ₱51.12, ₱48.85, and ₱53.80. a. What is the median value? b. What is the implication based on the median value? Solution: a. What is the median value? Step 1: Arrange the data in an ascending or a descending order. This is arranged in ascending order: 48.85, 49.69, 50.45, 50.5, 51.12, 52.8, 53.80, 54.56. Step 2: Determine the middlemost value. Since the number of values is 8, which is even, the median is the average of 50.5 and 51.12, which is 50.81. The median value of the oil prices is ₱50.81. 5.3. Interpreting Data: Measures of Central Tendency 11 Unit 5: Presentation and Analysis of Business Data b. What is the implication based on the median value? A median value of 50.81 means that 50% of the data are less than 50.81 while the other 50% are greater than 50.81. Grouped Data For grouped data, you can compute the median by following these steps. 1. Find the cumulative frequency. 2. Determine the median class. 3. When the median class is determined, identify the lower limit and frequency of the median class. Also, identify the cumulative frequency below the median class and the class size as well. 4. Substitute the values to the formula. The formula in solving the median of a grouped data is given by: where, x̄ = median Lm = lower limit of median class cfb = cumulative frequency below the median class n = number of values fm = frequency of the median class c = class size Properties of the median (x̃): 1. It may or may not be an actual score in the distribution. 2. It is not easily affected by outliers. 5.3. Interpreting Data: Measures of Central Tendency 12 Unit 5: Presentation and Analysis of Business Data Closer Look Example 1 The table below shows the ages of customers who visit the candy store. What is the median value? Age Frequency 3-9 11 10-16 4 17-23 2 24-30 2 31-37 1 n 20 Solution: 1. Find the cumulative frequency. Start adding from the frequency of the lowest score interval to the highest one. Age Frequency Cumulative Frequency 3-9 11 11 10-16 4 15 17-23 2 17 24-30 2 19 31-37 1 20 2. Determine the median class. In order to determine the median class, we use n/2, where n is the total number of data. 20/2 is 10 which is found in the first class. 5.3. Interpreting Data: Measures of Central Tendency 13 Unit 5: Presentation and Analysis of Business Data Age Frequency Cumulative Frequency 3-9 11 11 10-16 4 15 17-23 2 17 24-30 2 19 31-37 1 20 3. When the median class is determined, identify the lower limit and frequency of the median class. Also, identify the cumulative frequency below the median class and the class size as well. Lower limit (median class) = Lm - 0.5 = 3 - 0.5 = 2.5 Frequency (median class) = 11 Cumulative frequency (below the median class) = 0 (no class below the median class) Class size = 7 4. Substitute the values to the formula. The median value is 8.86 years old. 5.3. Interpreting Data: Measures of Central Tendency 14 Unit 5: Presentation and Analysis of Business Data Mode The mode is a score that appears the most in a given data set. By mere inspection, mode can be determined right away. This measure of central tendency is used if a specific number or numbers appear/s many times. Ungrouped Data To obtain the modal value of an ungrouped data, we simply determine the most frequently repeating value. Suppose in a data set containing an ungrouped list of 3, 4, 5, 5, 5, 6, 7, 8, 9, and 10, the mode is 5 since it is the most frequently occurring value. Closer Look Example 1 If the values are 5, 7, 10, 7, 5, 2, 6, 11, 7, 6, and 6, what is/are the modal value/s? Solution: 7 and 6 are repeated three times and are considered the most frequently occurring scores in the data set. Thus, 6 and 7 are the modal values. Example 2 A business owner tallies the ages of the customers for his different stores. Store A Store B Store C 45 25 55 24 24 54 24 24 49 20 24 48 20 18 44 20 18 40 5.3. Interpreting Data: Measures of Central Tendency 15 Unit 5: Presentation and Analysis of Business Data 16 17 38 12 15 35 10 10 30 7 9 20 a. What are the modal values for stores A, B, and C? b. What is the interpretation for the modal values in (a)? Solution: a. What are the modal values for stores A, B, and C? For Store A: 20 For Store B: 24 For Store C: None b. What is the interpretation for the modal values in (a)? For stores A and B, the most common ages of people that shop are 20 and 24, respectively. On the other hand, store C has no frequently appearing data observed in the set. Grouped Data For grouped data, we can determine the mode by following these steps. 1. Identify the modal class. 2. When the modal class is determined, identify the lower limit of the modal class, the difference between the frequency of the modal class and the frequency before/after the modal class, and the class size. 3. Substitute the values to the formula. The formula in solving for the mode of a grouped data is given by: 5.3. Interpreting Data: Measures of Central Tendency 16 Unit 5: Presentation and Analysis of Business Data where, x̂ = mode Lmo = lower limit of the modal class D1 = frequency of the modal class minus the frequency before the modal class D2 = frequency of the modal class minus the frequency after the modal class c = class size Properties of the mode (x̂): 1. It may or may not exist. 2. If it exists, it could be more than one. 3. It is not affected by outliers. Closer Look Example 1 The table below shows the ages of customers who visit the candy store. What is the modal value? Age Frequency 3-9 11 10-16 4 17-23 2 24-30 2 31-37 1 Solution: 1. Identify the modal class. To determine the modal class, we look at the class with the highest frequency. In this case, the class 3-9. Age Frequency 3-9 11 10-16 4 5.3. Interpreting Data: Measures of Central Tendency 17 Unit 5: Presentation and Analysis of Business Data 17-23 2 24-30 2 31-37 1 2. When the modal class is determined, identify the lower limit of the modal class, the difference between the frequency of the modal class and the frequency before/after the modal class, and the class size. Lower limit (modal class) = Lmo - 0.5 = 3 - 0.5 = 2.5 D1 = 11-0 = 11 D2 = 11-4 = 7 Class size = 7 3. Substitute the values to the formula. The modal value is 6.78 years old. Store A Store 5.3. Interpreting Data: Measures of Central Tendency 18 Unit 5: Presentation and Analysis of Business Data Check Your Progress Sheryl surveyed different room rates. She found out that she can rent a room with rates of ₱2,500 and ₱28,000. Some other room rates that she considered are ₱5,000, ₱7,500, ₱4,000, ₱6,000, ₱3,000, ₱7,000, and ₱5,500. What are the median and modal values? Business-Related Problems There are many business-related problems involving the concept of central tendency measures. We can solve them by following these steps: Step 1: Identify what is required in the problem. Step 2: Identify the given in the problem. Step 3: Write the working equation. Step 4: Substitute the given values. Step 5: Find the answer. 5.3. Interpreting Data: Measures of Central Tendency 19 Unit 5: Presentation and Analysis of Business Data Closer Look Example 1 XY Supermarket’s 30 regular employees receive an average monthly salary of ₱12,400. How much does XY Supermarket have to allot to the employees’ monthly salary? Solution: Step 1: Identify what is required in the problem. You are asked to determine the total money allotted for the employees’ salary every month (T). Step 2: Identify the given in the problem. 30 number of employees (n) ₱12,400 average monthly salary (x) Step 3: Write the working equation. Step 4: Substitute the given values. Step 5: Find the answer. Therefore, XY Supermarket has to allot ₱372,000 to the employees’ monthly salaries. 5.3. Interpreting Data: Measures of Central Tendency 20 Unit 5: Presentation and Analysis of Business Data Example 2 Dan works in a cell phone shop. To receive an incentive, he needs to sell an average of six cell phones daily for one week. He sold 3, 5, 4, and 9 devices for Monday, Tuesday, Wednesday, and Thursday, respectively. If he could sell 7 units for both Friday and Saturday, how many cell phones would he need to sell to receive an incentive? Solution: Step 1: Identify what is required in the problem. You are asked to determine the number of cell phone units (g) Dan needs to sell to meet the daily average target and receive an incentive. Step 2: Identify the given in the problem. 3 units Monday (a) 5 units Tuesday (b) 4 units Wednesday (c) 9 units Thursday (d) 7 units Friday (e) 7 units Saturday (f) Step 3: Write the working equation. Step 4: Substitute the given values. 5.3. Interpreting Data: Measures of Central Tendency 21 Unit 5: Presentation and Analysis of Business Data Step 5: Find the answer. Therefore, Dan needs to sell at least 7 units to meet the daily average target and receive an incentive. Check Your Progress Solve the following problems involving measures of central tendency. 1. Mr. Jimmy, a business owner, is confident that he will reach his first million sales for a whole month of effort by targeting an average daily income of ₱34,000 and more. Is Mr. Jimmy correct? (Use 1 month = 30 days) 5.3. Interpreting Data: Measures of Central Tendency 22 Unit 5: Presentation and Analysis of Business Data 2. During six days, AAA Store served an average of 180 customers per day. What number of clients do they need on the seventh day to reach their daily average of 190? Case Study The Mean Versus the Median: A Case Study in 4-R Act Litigation Recently, railroads asserted that their effective property tax rate should be equalized to the median of the rates charged to other taxpayers in a dispute brought under the "4-R Act." However, in the statute’s phrasing, it is mandated that a weighted mean be used in the calculation. No technical property or data nature dictates the choice between these two measures of location; instead, it is dictated by the statute that defines the objectives of statistical operations, which dictates the choice between these two measures of location. Before selecting an estimator, it is necessary to determine which parameter is being attempted to be estimated. Although this principle appears to be self-evident, unfortunate experience has shown that it is not. 5.3. Interpreting Data: Measures of Central Tendency 23 Unit 5: Presentation and Analysis of Business Data The Mean versus the Median: A Case Study in 4-R Act Litigation D. A. Freedman, “The Mean versus the Median: A Case Study in 4-R Act Litigation,” https://www.jstor.org/stable/1391684 October 22, 2013, last accessed on December 29, 2021. Keep in Mind A measure of central tendency refers to a single score or value that can best describe a given set of observations. The three forms of measures of central tendency are mean, median, or mode. The mean is the most stable measure of central tendency and can be obtained by determining the average of the set of scores. The median is the middlemost score of a set of distributions if arranged in an ascending or a descending order. The mode is a score that appears the most in a given data set and can be determined by mere inspection. Measure of How To Obtain How To Obtain Central (Ungrouped Data) (Grouped Data) Tendency Mean Median Middlemost value (data must be in order). Mode Most frequently occurring value. 5.3. Interpreting Data: Measures of Central Tendency 24 Unit 5: Presentation and Analysis of Business Data Try This A. Identification. Identify which measure of central tendency is described. Write in the blank space provided whether it is mean, median, or mode. ________________ 1. It is the most stable form of central tendency. ________________ 2. It is used when there are extreme values. ________________ 3. It is the most frequently repeating number. ________________ 4. It is a representative of all scores. ________________ 5. It is sometimes non-existent. B. True or False. Write true if the statement is correct. Otherwise, write false. ________________ 1. In computing the mean, we add all the values and divide the sum by the number of values present. ________________ 2. The computation of the mean of a grouped data involves the cumulative frequency. ________________ 3. Data must be in an ascending or a descending order when solving for the median. ________________ 4. The median value of an even data set is not possible. ________________ 5. Mode can have more than one value. 5.3. Interpreting Data: Measures of Central Tendency 25 Unit 5: Presentation and Analysis of Business Data Practice Your Skills Fact or Bluff Determine whether the given statement is a fact or a bluff by putting a checkmark (✓). Read and solve the given situation below. Pedro, a farmer, lists the number of sacks that his rice farm has harvested in ten years. These are 80, 85, 90, 82, 90, 100, 94, 110, 104, and 115. Statements Fact Bluff 1. Median is less than mode. 2. Mode is greater than 80. 3. Mode is less than mean. 4. What happens to the mean value if 106 is added? 5. How does a modal value change if another distinct number is added? 5.3. Interpreting Data: Measures of Central Tendency 26 Unit 5: Presentation and Analysis of Business Data Challenge Yourself Answer the following questions. 1. A journalist surveys to identify the median age of people who watch her documentaries. The ages are 21, 27, 6, 45, 48, 56, 72, 37, 12, 29, 15, 35, and 40. She concluded that the median age would increase from 35 to 36 years old if another person is added to the list, which is 70 years old. Is she correct? Support your answer. 2. Jerry is starting an online business, and he uses an online advertisement platform to reach more customers. For the first eight days, he got an average of 95 people every day. How many people do his ads need to reach on the ninth day to increase the average number of people reached to 158? 5.3. Interpreting Data: Measures of Central Tendency 27 Unit 5: Presentation and Analysis of Business Data 3. Norman sells different types of pastries. For 5 days, he recorded an average of ₱3,550 daily income. How much does he need to achieve on the sixth day to increase his average daily income to ₱3,625? 5.3. Interpreting Data: Measures of Central Tendency 28 Unit 5: Presentation and Analysis of Business Data Photo Credit Peso Money Currency Philippines by florantevaldez is free to use under thePixabay License via Pixabay. Bibliography Abao, Zenon R., Eugenio S. Guhao Jr., Jose B. Maribbay, Roland S. Zorilla, Violeta C. Mendoza, Jesus P. Mercado, Beda H. Essler, Fe G. Partible, and Aurora E. V. Business Mathematics. Mandaluyong City: Books Atpb. Publishing Corp, 2008. Clendenen, Gary, and Stanley Salzman. Business Mathematics. Essex: Pearson Education Limited, 2015. Freedman, D. A. “The Mean versus the Median: A Case Study in 4-R Act Litigation.” Journal of Business & Economic Statistics 3, no. 1 (1985): 1–13. https://doi.org/10.2307/1391684. Villanueva, T. T. Business Mathematics. Valenzuela City: Tru-Copy Publishing House, Inc., 2017. 5.3. Interpreting Data: Measures of Central Tendency 29